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The aim of this work is to point out a strong structural phenomenon hiddenbehind the existence of normalizers through the investigation of this propertyin the non-pointed context: given any category E, a certain property of thefibration of points: Pt(E) --> E guarentees the existence of normalizers. Thisproperty becomes a characterization of this existence when E is quasi-pointedand protomodular. This property is also showed to be equivalent to a propertyof the category GrdE of internal groupoids in E which is a kind of opposite,for the monomorphic internal functors, of the comprehensive factorization.

Normalizers in the non-pointed context: a weak case of extremal decomposition